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PMath 467/667 Homework 7 Solutions
1. [Problem 12-6 from the textbook] Let E be the complement of the diagonal
in R3 R3 that is, let E = cfw_(v, w) R3 R3 | v = w. Let be the
equivalence relation (x, y) (y, x). Compute the fundamental group of
E/ .
Solutio

PMath 467/667 Homework 8 Solutions
1. Compute Hp (X) for all integers p 0, where X is the wedge product of
S 1 with S 2 .
Solution: Since X is connected, we have H0 (X) Z. The construction
=
2
1
of X from S and S can be viewed as the attaching of a 2-cell

PMath 467/667 Homework 3 Solutions
1. Let X be a Hausdor, connected, two-dimensional cell complex with a
nite number of cells and no 1-cells. Show that X is homeomorphic to the
wedge product of a nite number of copies of S 2 . That is, X is homeomorphic t

PMath 467/667 Homework 2 Solutions
1. (Related to problem 3-24 from the textbook) Consider the action of S 1 on
R2 by rotation that is, the complex number ei corresponds to a rotation
by . Prove that the quotient of R2 by this action is homeomorphic to [0

PMath 467/667 Homework 4 Solutions
1. Let X be the quotient of an octagon O by identifying two pairs of opposite
sides, as in the following picture:
Prove that X is a 2-manifold with boundary.
Solution: It is clear that X is a second countable Hausdor spa

PMath 467/667 Homework 6 Solutions
1. Let X be homeomorphic to the connected sum of k tori, where k 1 is a
positive integer. Let f : S 1 S 1 X be a covering map with nitely many
sheets. Prove that k = 1.
Solution: Let P be any point on S 1 S 1 . The monod

PMath 467/667 Homework 5 Solutions
1. Let X = R3 cfw_x = y = 0 be the complement of the z-axis in R3 . Prove
that X is homotopy equivalent to S 2 cfw_P, Q, where P and Q are two points.
Solution: First, notice that S 2 cfw_P1 , Q1 is homeomorphic to S 2

PMath 467/667 Homework 1 Solutions
1. (Part of problem 3-4 from the textbook) Show that every closed ball in
Rn in an n-dimensional manifold with boundary.
Solution: First, its convenient to note that every closed ball in Rn is homeomorphic to every other